Force plate calibration
WCR 20131104
A force plate converts physical forces in three dimensions (Fx, Fy, Fz) and moments about
three axes (Mx, My, Mz) to voltages. The voltages can be measured with a analog-to-digital
converter (ADC) and saved. The force plate has six voltage outputs that are supposed to
correspond to the six forces and moments. The voltages are labelled Vfx, Vfy, Vfz, Vmx, Vmy,
and Vmz. An ideal force plate would have the following relationship between physical forces
and moments and voltages:1
𝑉𝑓𝑥 = 𝑆1 𝐹𝑥
(1a)
𝑉𝑓𝑦 = 𝑆2 𝐹𝑦
(1b)
𝑉𝑓𝑧 = 𝑆3 𝐹𝑧
(1c)
𝑉𝑚𝑥 = 𝑆4 𝑀𝑥
(1d)
𝑉𝑚𝑦 = 𝑆5 𝑀𝑦
(1e)
𝑉𝑚𝑧 = 𝑆6 𝑀𝑧
(1f)
The quantities S1 through S6 are the sensitivities of the forceplate transducers to forces and
moments. Due to unavoidable engineering imperfections, each of the six output voltages is
slightly affected by other forces and moments. For example, voltage Vfx, which should be
sensitive to Fx and nothing else, may actually be something like the following:
𝑉𝑓𝑥 = 10𝐹𝑥 − 0.06𝐹𝑦 + 0.08𝐹𝑧 − 0.02𝑀𝑥 − 0.04𝑀𝑦 + 0.05𝑀𝑧
This “contamination” of the output voltages by unwanted forces and moments is sometimes
called crosstalk. The sensitivities, are different for each voltage and each force and moment.
The following equations summarize the response of the force plate.
𝑉𝑓𝑥 = 𝑆11 𝐹𝑥 + 𝑆12 𝐹𝑦 + 𝑆13 𝐹𝑧 + 𝑆14 𝑀𝑥 + 𝑆15 𝑀𝑦 + 𝑆16 𝑀𝑧
(2a)
𝑉𝑓𝑦 = 𝑆21 𝐹𝑥 + 𝑆22 𝐹𝑦 + 𝑆23 𝐹𝑧 + 𝑆24 𝑀𝑥 + 𝑆25 𝑀𝑦 + 𝑆26 𝑀𝑧
(2b)
𝑉𝑓𝑧 = 𝑆31 𝐹𝑥 + 𝑆32 𝐹𝑦 + 𝑆33 𝐹𝑧 + 𝑆34 𝑀𝑥 + 𝑆35 𝑀𝑦 + 𝑆36 𝑀𝑧
(2c)
𝑉𝑚𝑥 = 𝑆41 𝐹𝑥 + 𝑆42 𝐹𝑦 + 𝑆43 𝐹𝑧 + 𝑆44 𝑀𝑥 + 𝑆45 𝑀𝑦 + 𝑆46 𝑀𝑧
(2d)
𝑉𝑚𝑦 = 𝑆51 𝐹𝑥 + 𝑆52 𝐹𝑦 + 𝑆53 𝐹𝑧 + 𝑆54 𝑀𝑥 + 𝑆55 𝑀𝑦 + 𝑆56 𝑀𝑧
(2e)
𝑉𝑚𝑧 = 𝑆61 𝐹𝑥 + 𝑆62 𝐹𝑦 + 𝑆63 𝐹𝑧 + 𝑆64 𝑀𝑥 + 𝑆65 𝑀𝑦 + 𝑆66 𝑀𝑧
(2f)
The above set of equations can be written as a single equation using vectors and a matrix:
𝑉𝑓𝑥
𝑆11
𝑉𝑓𝑦
𝑆21
𝑆31
𝑉𝑓𝑧
=
𝑆41
𝑉𝑚𝑥
𝑆51
𝑉𝑚𝑦
( 𝑉𝑚𝑧 ) (𝑆61
𝑆12
𝑆22
𝑆32
𝑆42
𝑆52
𝑆62
𝑆13
𝑆23
𝑆33
𝑆43
𝑆53
𝑆63
𝑆14
𝑆24
𝑆34
𝑆44
𝑆54
𝑆64
𝑆15
𝑆25
𝑆35
𝑆45
𝑆55
𝑆65
𝐹𝑥
𝑆16
𝐹𝑦
𝑆26
𝑆36
𝐹𝑧
𝑀𝑥
𝑆46
𝑀𝑦
𝑆56
𝑆66 ) ( 𝑀𝑧 )
(3)
or simply
𝑽 = 𝑺𝑿
where V is the vector of voltages, S is the sensitivity matrix, and X is the vector of physical
quantities (forces and moments).
We assume the force plate voltage outputs have been properly “zeroed”, i.e. adjusted so that the voltage is zero
when the force or moment is zero. We also assume the force plate has a linear response.
1
(4)
We measure voltages and we wish to recover the physical quantities. In other words, we
wish to determine the vector X, based on our measurement of V. Using the rules of matrix
arithmetic, we pre-multiply by the inverse of S:
𝑺−𝟏 𝑽 = 𝑺−𝟏 𝑺𝑿
(5)
𝑺−𝟏 𝑽 = 𝑰𝑿
(6)
𝑿 = 𝑺−𝟏 𝑽
(7)
Force plate manufacturer AMTI provides factory calibration information: the 6x6 sensitivity
matrix S and the inverted sensitivity matrix S-1, which AMTI calls B.
The elements of sensitivity matrix S have units of V/N (columns 1-3) or V/(N-m) (columns
4-6).2 The elements of inverted sensitivity matrix B have units of N/V (rows 1-3) and N-m/V
(rows 4-6). Substituting B for S-1, we can rewrite (7) as follows:
𝐹𝑥
𝐵11
𝐹𝑦
𝐵21
𝐵31
𝐹𝑧
=
𝑀𝑥
𝐵41
𝑀𝑦
𝐵51
( 𝑀𝑧 ) (𝐵61
𝐵12
𝐵22
𝐵32
𝐵42
𝐵52
𝐵62
𝐵13
𝐵23
𝐵33
𝐵43
𝐵53
𝐵63
𝐵14
𝐵24
𝐵34
𝐵44
𝐵54
𝐵64
𝐵15
𝐵25
𝐵35
𝐵45
𝐵55
𝐵65
𝑉𝑓𝑥
𝐵16
𝑉𝑓𝑦
𝐵26
𝐵36
𝑉𝑓𝑧
𝐵46
𝑉𝑚𝑥
𝐵56
𝑉𝑚𝑦
𝐵66 ) ( 𝑉𝑚𝑧 )
(8)
Equation 8 tells us how to undo the effects of crosstalk and obtain physical quantities (forces
and moments) from the measured voltages.
Crosstalk: A Simple Example
Suppose we measure EMG and EKG simultaneously. The EMG is collected from just below
the xiphoid, in order to measure diaphragm activity (breathing). A lead II EKG is also collected
(LL-RA). The actual voltages due to EMG and EKG are EMG(t) and EKG(t). However, our
measuring electrodes pick up some of the other signal, so our measured voltages are
𝑉1 = 𝐸𝑀𝐺 − 0.2 ∗ 𝐸𝐾𝐺
𝑉2 = 𝐸𝐾𝐺 + 0.2 ∗ 𝐸𝑀𝐺
We want to extract the true EMG and EKG. We do it by solving the system of 2 equations
and 2 unknowns (EMG and EKG).
𝑉1 + 0.2 ∗ 𝑉2 = 𝐸𝑀𝐺 − 0.2 ∗ 𝐸𝐾𝐺 + 0.2 ∗ (𝐸𝐾𝐺 + 0.2 ∗ 𝐸𝑀𝐺)
𝑉1 + 0.2 ∗ 𝑉2 = 𝐸𝑀𝐺 − 0.2 ∗ 𝐸𝐾𝐺 + 0.2 ∗ 𝐸𝐾𝐺 + 0.04 ∗ 𝐸𝑀𝐺
𝑉1 + 0.2 ∗ 𝑉2 = 1.04 ∗ 𝐸𝑀𝐺
(𝑉1 + 0.2 ∗ 𝑉2 )
𝐸𝑀𝐺 =
= 0.9615𝑉1 + 0.1923𝑉2
1.04
By a similar method we also find that
2
The matrix S provided by AMTI has units of of µV/(Vex-N) (columns 1-3) or µV/(Vex-N-m) (columns 4-6).
We multiply the AMTI-specified matrix values by the excitation voltage used (Vex=10V for each channel, in the
Biomechanics Lab in Rust 210) and by the voltage amplifier gain (gain=1000 in the Biomechanics Laboratory). The
net effect of excitation voltage and amplifier gain is that the elements of S are multiplied by 10,000 to get actual
µV/N or µV/(N-m). Alternatively, divide each element of S by 100 to get units of V/N or V/(N-m).
The inverted sensitivity matrix B provided by AMTI should be multiplied by 100 to give elements with units of
N/V (rows 1-3) and N-m/V (rows 4-6). The text files forceplate[n]_inv_sens_mtx.txt contain the AMTI values
multiplied by 100.
The AMTI inverted sensitivity matrix (B) is not the exact inverse of the AMTI sensitivity matrix (S). This may
be due to rounding. I tried to find a pair of matrices that are more nearly inverse and are the same as S and B to
within rounding error of the printed factory values. See finetune_invmatrices.vi for how this was done. The
resulting matrices are forceplate4_sens_mtx_corr.txt and forceplate4_inv_sens_mtx_corr.txt.
(𝑉2 − 0.2 ∗ 𝑉1 )
= −0.1923𝑉1 + 0.9615𝑉2
1.04
We can extract the true EMG and EKG using the equations above.
We can also see that these equations can be represented with matrices. If X=(EMG,EKG) and
V=(V1,V2), then
𝑉
𝑆
𝑆12 𝐸𝑀𝐺
( 1 ) = ( 11
)(
)
𝑉2
𝑆21 𝑆22 𝐸𝐾𝐺
Where
𝑆
𝑆12
1.0 −0.2
𝑆 = ( 11
)=(
)
𝑆21 𝑆22
0.2 1.0
Then we can define B=S-1 and obtain
𝑆
𝑆12 −1 𝑉1
𝑆
𝑆12 −1 𝑆11 𝑆12 𝐸𝑀𝐺
( 11
) ( ) = ( 11
) (
)(
)
𝑉2
𝑆21 𝑆22
𝑆21 𝑆22
𝑆21 𝑆22 𝐸𝐾𝐺
𝐸𝐾𝐺 =
which simplifies to
(
𝑆
𝐸𝑀𝐺
) = ( 11
𝑆21
𝐸𝐾𝐺
𝑆12 −1 𝑉1
) ( )
𝑉2
𝑆22
We define B to be the inverse of S: B=S-1 .
𝑆
𝑆12 −1
0.9615 0.1923
𝐵 = ( 11
) =(
)
𝑆21 𝑆22
−0.1923 0.9613
Then
𝐸𝑀𝐺
0.9615 0.1923 𝑉1
(
)=(
)( )
𝐸𝐾𝐺
−0.1923 0.9613 𝑉2